Long cylindrical cores made of plastic or spirally wound paperboard are commonly used to wind large quantities of web material such as, for example, paper or film into rolls for storage and transport. Some cores have inner diameters (ID) that are larger than those of other cores. For example, cores having IDs of 150 millimeter (mm) are common as are cores having 76 mm IDs. It is desirable to mount both large and small ID cores on winding machines such as double drum winders that have spindles or chucks configured to accept smaller ID cores only. In order to do this, core adapters may be installed in the ends of the larger ID cores and the adapters have central bores that can be mounted on the smaller ID spindles or core chucks of a winding or other machine. Traditional core adapters take many forms such as, for instance, leaf adapters with leafs that can expand to lock the adapter in the end of the core, rubber air or pneumatic adapters that are inserted in the core ends and inflated to lock them in place, and others. Core adapters made of wound paper in the form of one or multiple concentrically arranged components also are known. While somewhat successful, these traditional adapters can be expensive, do not always ensure a precisely centered smaller central opening, and can be unintentionally left off, which necessitates a time consuming rewinding of the web material. Traditional core adapters also may not ensure precise concentricity of the smaller opening of the adapter with the larger opening of the core. It is to a core adapter that addresses these and other shortcomings of traditional core adapters that the present invention is primarily directed.